Finish removing the BigInts from * for FD{Int128}!

Finally implements the fast-multiplication optimization from
#45, but this
time for 128-bit FixedDecimals! :)

This is a follow-up to
#93, which
introduces an Int256 type for widemul. However, the fldmod still
required 2 BigInt allocations.

Now, this PR uses a custom implementation of the LLVM div-by-const
optimization for (U)Int256, which briefly widens to Int512 (😅) to
perform the fldmod by the constant 10^f coefficient.

This brings 128-bit FD multiply to the same performance as 64-bit. :)

Author: NHDaly

src/FixedPointDecimals.jl

@@ -36,9 +36,11 @@ export checked_abs, checked_add, checked_cld, checked_div, checked_fld,
 
 using Base: decompose, BitInteger
 
-import BitIntegers  # For 128-bit _widemul / _widen
+using BitIntegers: BitIntegers, UInt256, Int256
 import Parsers
 
+include("fldmod-by-const.jl")
+
 # floats that support fma and are roughly IEEE-like
 const FMAFloat = Union{Float16, Float32, Float64, BigFloat}
 
@@ -129,8 +131,10 @@ _widemul(x::Unsigned,y::Signed) = signed(_widen(x)) * _widen(y)
 
 # Custom widen implementation to avoid the cost of widening to BigInt.
 # FD{Int128} operations should widen to 256 bits internally, rather than to a BigInt.
-_widen(::Type{Int128}) = BitIntegers.Int256
-_widen(::Type{UInt128}) = BitIntegers.UInt256
+_widen(::Type{Int128}) = Int256
+_widen(::Type{UInt128}) = UInt256
+_widen(::Type{Int256}) = BitIntegers.Int512
+_widen(::Type{UInt256}) = BitIntegers.UInt512
 _widen(t::Type) = widen(t)
 _widen(x::T) where {T} = (_widen(T))(x)
 
@@ -196,18 +200,12 @@ function _round_to_nearest(quotient::T,
 end
 _round_to_nearest(q, r, d, m=RoundNearest) = _round_to_nearest(promote(q, r, d)..., m)
 
-# In many of our calls to fldmod, `y` is a constant (the coefficient, 10^f). However, since
-# `fldmod` is sometimes not being inlined, that constant information is not available to the
-# optimizer. We need an inlined version of fldmod so that the compiler can replace expensive
-# divide-by-power-of-ten instructions with the cheaper multiply-by-inverse-coefficient.
-@inline fldmodinline(x,y) = (fld(x,y), mod(x,y))
-
 # multiplication rounds to nearest even representation
 # TODO: can we use floating point to speed this up? after we build a
 # correctness test suite.
 function Base.:*(x::FD{T, f}, y::FD{T, f}) where {T, f}
     powt = coefficient(FD{T, f})
-    quotient, remainder = fldmodinline(_widemul(x.i, y.i), powt)
+    quotient, remainder = fldmod_by_const(_widemul(x.i, y.i), Val(powt))
     reinterpret(FD{T, f}, _round_to_nearest(quotient, remainder, powt))
 end
 
@@ -234,12 +232,12 @@ function Base.round(x::FD{T, f},
                              RoundingMode{:NearestTiesUp},
                              RoundingMode{:NearestTiesAway}}=RoundNearest) where {T, f}
     powt = coefficient(FD{T, f})
-    quotient, remainder = fldmodinline(x.i, powt)
+    quotient, remainder = fldmod_by_const(x.i, Val(powt))
     FD{T, f}(_round_to_nearest(quotient, remainder, powt, m))
 end
 function Base.ceil(x::FD{T, f}) where {T, f}
     powt = coefficient(FD{T, f})
-    quotient, remainder = fldmodinline(x.i, powt)
+    quotient, remainder = fldmod_by_const(x.i, Val(powt))
     if remainder > 0
         FD{T, f}(quotient + one(quotient))
     else
@@ -435,7 +433,7 @@ function Base.checked_sub(x::T, y::T) where {T<:FD}
 end
 function Base.checked_mul(x::FD{T,f}, y::FD{T,f}) where {T<:Integer,f}
     powt = coefficient(FD{T, f})
-    quotient, remainder = fldmodinline(_widemul(x.i, y.i), powt)
+    quotient, remainder = fldmod_by_const(_widemul(x.i, y.i), Val(powt))
     v = _round_to_nearest(quotient, remainder, powt)
     typemin(T) <= v <= typemax(T) || Base.Checked.throw_overflowerr_binaryop(:*, x, y)
     return reinterpret(FD{T, f}, T(v))

test/fldmod-by-const_tests.jl

@@ -0,0 +1,56 @@
+
+@testset "calculate_inverse_coeff signed" begin
+    using FixedPointDecimals: calculate_inverse_coeff
+
+    # The correct magic number here comes from investigating the following native code
+    # produced on an m2 aarch64 macbook pro:
+    # @code_native ((x)->fld(x,100))(2)
+    #   ...
+    #         mov     x8, #55051
+    #         movk    x8, #28835, lsl #16
+    #         movk    x8, #2621, lsl #32
+    #         movk    x8, #41943, lsl #48
+    # Where:
+    # julia> 55051 | 28835 << 16 | 2621 << 32 | 41943 << 48
+    # -6640827866535438581
+    @test calculate_inverse_coeff(Int64, 100) == (-6640827866535438581, 6)
+
+    # Same for the tests below:
+
+    # (LLVM's magic number is shifted one bit less, then they shift by 2, instead of 3,
+    #  but the result is the same.)
+    @test calculate_inverse_coeff(Int64, 10) == (7378697629483820647 << 1, 3)
+
+    @test calculate_inverse_coeff(Int64, 1) == (1, 0)
+end
+
+@testset "calculate_inverse_coeff signed 4" begin
+    using FixedPointDecimals: calculate_inverse_coeff
+
+    # Same here, our magic number is shifted 2 bits more than LLVM's
+    @test calculate_inverse_coeff(UInt64, 100) == (0xa3d70a3d70a3d70b, 6)
+
+    @test calculate_inverse_coeff(UInt64, 1) == (UInt64(0x1), 0)
+end
+
+@testset "div_by_const" begin
+    vals = [2432, 100, 0x1, Int32(10000), typemax(Int64), typemax(Int16), 8, Int64(2)^32]
+    for a_base in vals
+        # Only test negative numbers on `a`, since div_by_const requires b > 0.
+        @testset for (a, b, f) in Iterators.product((a_base, -a_base), vals, (unsigned, signed))
+            a, b = promote(f(a), f(b))
+            @test FixedPointDecimals.div_by_const(a, Val(b)) == a ÷ b
+        end
+    end
+end
+
+@testset "fldmod_by_const" begin
+    vals = [2432, 100, 0x1, Int32(10000), typemax(Int64), typemax(Int16), 8, Int64(2)^32]
+    for a_base in vals
+        # Only test negative numbers on `a`, since fldmod_by_const requires b > 0.
+        @testset for (a, b, f) in Iterators.product((a_base, -a_base), vals, (unsigned, signed))
+            a, b = promote(f(a), f(b))
+            @test FixedPointDecimals.fldmod_by_const(a, Val(b)) == fldmod(a, b)
+        end
+    end
+end

test/runtests.jl

@@ -10,4 +10,9 @@ include(joinpath(pkg_path, "test", "utils.jl"))
 
 @testset "FixedPointDecimals" begin
     include("FixedDecimal.jl")
-end  # global testset
+end
+
+@testset "FixedPointDecimals" begin
+    include("fldmod-by-const_tests.jl")
+end
+